A novel extension of Independent Component and Independent Vector Analysis for blind extraction/separation of one or several sources from time-varying mixtures is proposed. The mixtures are assumed to be separable source-by-source in series or in parallel based on a recently proposed mixing model that allows for the movements of the desired source while the separating beamformer is time-invariant. The popular FastICA algorithm is extended for these mixtures in one-unit, symmetric and block-deflation variants. The algorithms are derived within a unified framework so that they are applicable in the real-valued as well as complex-valued domains, and jointly to several mixtures, similar to Independent Vector Analysis. Performance analysis of the one-unit algorithm is provided; it shows its asymptotic efficiency under the given mixing and statistical models. Numerical simulations corroborate the validity of the analysis, confirm the usefulness of the algorithms in separation of moving sources, and show the superior speed of convergence and ability to separate super-Gaussian as well as sub-Gaussian signals.